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In this article, we study the resolvent energy and the pseudospectrum energy of a sequence C12n of fullerenes with exactly 12n carbon atoms. In particular, the resolvent energy of these fullerenes ...

A (k,6)-fullerene graph is a planar 3-connected cubic graph whose faces are k-gons and hexagons. The aim of this paper is to compute the equitable chromatic, b-chromatic and list chromatic numbers ...

Suppose G is a graph, A(G) its adjacency matrix, and μ1(G)≤(G)μ2(G)≤ ... ≤ μ(n)(G)are eigenvalues of A(G). The numbers S(k)(G) = Σ(i) n = 1 μ(i)k (G), 0 ≤ k ≤ n -1 are said to be the k-th spectral moment of G and the sequence S(G) = (S0(G), S1 (G),..., S(n-1)(G)is called the spectral moments sequence of G. Suppose G1 and G2 are graphs. If there exists an integer k, 1 ≤ k ≤ n - 1, such that for each i, 0 ≤ i ≤ k - 1, S(i) (G1) = S(i)(G2) and S(k)(G1) < S(k)(G2) then we write G1

A molecular graph is a graph in which vertices are atoms and edges are chemical bonds. In this paper, we describe a method for computing the symmetry of some giant carbon molecules. Some conjectures and open questions are also presented.

The Timisoara-eccentricity (TM-EC) index of a molecular graphis defined as the sum of  i i i over all at- oms i in � , where  i,  i and  i are the degree, eccentricity and the number of atoms at distance  i from atom i. The topo- logical efficiency index ofis defined as = 2W / Nw , where W denotes the Wiener index, w is the minimal vertex contribution and N is the number of carbon atoms. This paper is devoted to the study of nanocones and fullerenes by these new graph in- variants. It is proved that the TM-EC index of a fullerenecan be bounded by a polynomial of degree 2, for twelve infinite series of fullerenes. It is also shown that in one pentagonal carbon nanocone with exactly 5n 2 + 10n + 5 carbon atoms, we have   1.24 and TM - EC = 280n 3 + 385n 2 + 195n + 40. Finally, we examine the dual of this nanocone and prove that we have   1.24 and TM - EC = 70n 3 + 20n 2 - 5n.

A (3,6)-fullerene is a 3-connected cubic planar graph whose faces are triangles and hexagons. In this paper, it is proved that for a (3,6)-fullerene graph F, |Aut(F)| divides 24. An open question is also presented.

A molecular graph M is a simple graph in which atoms and chemical bonds are the vertices and edges of M, respectively. The molecular graph M is called a fullerene graph, if M is the molecular graph of a fullerene molecule. It is well-known that such molecules exist for even integers n ≥ 24 or n = 20. The aim of this paper is to investigate the topological properties of a class of fullerene molecules containing 60 + 12n carbon atoms.

Fullerenes are carbon cage molecules having 12 pentagonal and (n/2 – 10) hexagonal faces, where 20 ≤ n (≠ 22) is an even integer. In this paper, an infinite class of fullerenes with 10n carbon atoms is investigated. We prove that the vertex–PI, Szeged and revised Szeged indices of this family are computed by formulas 150n2 − 100n, 250n3 + 3075n − 13800 and 250n3 + 250n2 + 4275n − 15650, respectively, when n > 10 is a positive integer. A MATLAB program is also presented that is useful for our calculations.

The energy of a graph G is a graph invariant defined as the summation of the absolute values of all eigenvalues of G . In this work the energy of a nanotorus and that of a (3,6)-fullerene are computed by using a result of John and Sachs [P.E. John, H. Sachs, Spectra of Toroidal graphs, Discrete Math. 309 (2009) 2663–2681].

A molecular graph G is said to be three connected if there does not exist a two-element set of vertices whose removal disconnects the graph. A three-connected cubic planar graph G = (V;E) is called m-generalized fullerene if it has two m-gons and all other faces are pentagons and hexagons. The aim of this chapter is to compute some distance topology of eight infinite sequences of 3-generalized fullerene.

A (3,6)-fullerene is a planar 3-connected cubic graph whose faces are triangles and hexagons. In this chapter, the modified Wiener and hyper–Wiener indices of three infinite classes of (3,6)-fullerenes are considered into account. Some open questions are also presented.

A molecular graph is a graph in which vertices are atoms and edges are chemical bonds. A graph is called 3-connected, if there does not exist two vertices whose removal disconnects the graph. A fullerene graph is a cubic, planar, and 3-connected whose faces are all pentagons or hexagons. A fullerene is a molecule, that is, its molecular graph is a fullerene graph. In 1991, Graovac and Pisanski (1991) proposed an algebraic modification of the Wiener index of a graph by considering its automorphism group. In this chapter, an algebraic modification for the hyper–Wiener index is presented. These quantities will be computed for some classes of fullerenes.

Suppose G is a molecular graph and Γ is a subgroup of its symmetry group. The Graovac-Pisanski index (GP index in short) of G with respect to Γ is a symmetry-based topological index. This topological index gives a distance-symmetry description of the molecular graph G. The aim of this paper is to compute the GP index of some classes of fullerenes and fullerene-like molecules.

Fullerene graphs are 3-regular planar graphs whose faces are pentagons or hexagons. In this paper, we first compute the number of paths of low order in a fullerene. Then apply these numbers to obtain the number of independent sets of size k and k -matchings in fullerenes, when k = 1 , 2 , 3 , 4 .

The Wiener polarity index W p ( G ) of a molecular graph G of order n is the number of unordered pairs of vertices u , v of G such that the distance d G ( u , v ) between u and v is 3. In this note, it is proved that in a triangle- and quadrangle-free connected graph G with the property that the cycles of G have at most one common edge, W p ( G ) = M 2 ( G ) − M 1 ( G ) − 5 N p − 3 N h + | E ( G ) | , where M 1 ( G ) , M 2 ( G ) , N p and N h denoted the first Zagreb index, the second Zagreb index, the number of pentagons and the number of hexagons, respectively. As a special case, it is proved that the Wiener polarity index of fullerenes with n carbon atoms is ( 9 n − 60 ) / 2 . The extremal values of catacondensed hexagonal systems, hexagonal cacti and polyphenylene chains with respect to the Wiener polarity index are also computed.

In this paper the Szeged and revised Szeged indices of an infinite family of IPR fullerenes with 50 + 10n carbon atoms and D 5h point group symmetry are computed. It seems that some chemico-physical properties of these fullerenes are derived from the mentioned topological invariants.

The modified eccentric connectivity polynomial of a molecular graph, G, is defined as \(\Lambda (G,x) = \Sigma ^a \in V(G)^n G(a)x^{\varepsilon _G (a)} \) where eG(a) is the eccentricity of vertex a and nG(a) is the sum of the degrees of its neighborhoods. In this paper, the polynomial for three infinite classes of fullerenes is computed.

In this article, the PI and Omega polynomials of an infinite family of IPR fullerenes are computed. A GAP program is presented that is useful for computing PI and omega polynomials of fullerenes.

Let G be a graph, e = uv ∈ E(G), mu(e) be the number of edges of G lying closer to u than to v and mv(e) be the number of edges of G lying closer to v than to u. Then the PI and edge Szeged polynomials of the graph G are defined as and , respectively. Here, mu(e) is the number of edges of G lying closer to u than to v and mv(e) is the number of edges of G lying closer to v than to u. These counting polynomials for an infinite family of fullerenes are computed.

Let G be a graph, e = uv ∈ E( G ), nu(e) be the number of vertices of G lying closer to u than to v and nv(e) be the number of vertices of G lying closer to v than to u. The vertex PI and Szeged polynomials of the graph G are defined as PI v( G ,x) = ∑e = uv xnu (e) + nv (e) and Sz ( G ,x) = ∑e = uv xnu (e)nv (e), respectively. In this paper, these counting polynomials for an infinite family of IPR fullerenes are computed.

In this paper, a new algorithm is presented which is useful for computing the automorphism group of chemical graphs. We compare our algorithm with those of Druffel, Schmidt and Wang. It is proved that the running time of the present algorithm is better than the mentioned algorithms. Also, the automorphism group of Euclidean graph of isomers for the fullerenes C180, C240, C260, C320, C500 and C720 are computed.

In this paper we, present some MATLAB and GAP programs and use them to find the automorphism group of the Euclidean graph of the C80 fullerence with connectivity and geometry of I h symmetry point group. It is proved that this group has order 120 and is isomorphic to I h ≊Z2×A5, where Z2 is, a cyclic group of order 2 and A5 is the alternating group on five symbols.

It is well known to associate an Euclidean graph to a molecule. Balasubramanian computed the Euclidean graphs and their automorphism groups for benzene, eclipsed and staggered forms of ethane and eclipsed and staggered forms of ferrocene [see Chem. Phys. Lett. 232 (1995), 415]. In this Letter, we proved an algorithm, which is useful for finding symmetry of molecules. Using this algorithm, a new simple method is described, by means of which it is possible to calculate the automorphism group of weighted graphs. We apply this method to compute the symmetry of the smallest fullerene C 20 . Finally, an upper bound for the order of automorphism group of weighted graphs is obtained.

Suppose G is a graph with V(G) = {v1, v2, ..., vn}. The eccentricity sequence of G, denoted by e(G), is the set of all e(vi). In this paper, the eccentricity sequences of two infinite classes of fullerenes are computed. As a consequence, the eccentric connectivity indices of these fullerenes are computed.

Suppose G is a graph and λ 1 , λ 2 , … λ n are the eigenvalues of G . The Estrada index E E ( G ) of G is defined as the sum of the terms e λ i , 1 ≤ i ≤ n . In this work some upper and lower bounds for the Estrada index of ( 4 , 6 ) -fullerene graphs are presented.

A (3,6)–fullerene is a cubic plane graph whose faces (including the outer face) have sizes 3 or 6. (4,6)–Fullerene graphs are defined analogously by interchanging triangles with quadrangles. (3,6)–Fullerenes have exactly four triangles and (4,6)–fullerenes have exactly 6 quadrangles. The (4,6)–fullerenes are also called boron fullerenes. In this chapter some infinite families of (3,6)–and (4,6)–fullerenes are presented. The modeling of these fullerenes by considering some topological indices is the main part of this chapter. Finally, some open questions are presented.

A molecular graph is a graph such that its vertices correspond to the atoms and the edges to the bonds of a given molecule. Fullerenes are molecules in the form of polyhedral closed cages made up entirely of n three-coordinate carbon atoms and having 12 pentagonal and (n/2–10) hexagonal faces, where n is equal or greater than 20. The molecular graph of a fullerene is called fullerene graph. In this chapter, the fullerene graphs under two new distance-based topological indices are investigated. Some open questions are also presented.

Helical Wrapping of Graphene sheets and their Self-assemble into Core-Shelled Composite Nanostructures with metallic particles.- First-Principles Study of the Electronic and Magnetic Properties of Defects in Carbon Nanostructures.- Structural defects on the Electronic Transport Properties of Carbon-based Nanostructures.- Topological versus Physical and Chemical Properties of Negatively Curved Carbon Surfaces.- Topochemistry of spatially extended sp2 nanocarbons: fullerenes, nanotubes, and grapheme.- A Pariser-Parr-Pople Model Hamiltonian based approach to the eletronic structure and optical properties of graphene nanostructures.- Topological invariants of Mobius-like graphenic nanostructures.- Spanning Fullerenes as Units in Crystal Networks.- Introducing "Colored" Molecular Topology by Reactivity Indices of Electronegativity and Chemical Hardness.- Nanostructures and Eigenvectors of Matrices.- Theoretical analysis of the reactivity of carbon nanotubes: local versus topological effects.- Computation of the Szeged index of some nanotubes and dendrimers.- The Edge-Wiener index and its computation for some nanostructures.- Study of Fullerenes by Some New Topological Index.- Topological Study of (3,6)- and (4,6)-Fullerenes.- Enumeration of Hetero - Molecules by Using Polya Theorem.

A topological index for a molecular graph is a number correlated some physic-chemical properties of the molecule under consideration. The Wiener index is the first reported distance based topological index applicable in chemistry. It is defined as half sum of the distances between all the pairs of vertices in a molecular graph. In this survey article a brief account on the development of the Wiener in the fascinating field of nanoscience are discussed

A counting polynomial C(G,x) is a sequence description of a topological property so that the exponents express the extent of its partitions while the coefficients are related to the occurrence of these partitions. Basic definitions and properties of the Omega polynomial Ω(G,x) and Sadhana polynomial Sd(G,x) are presented. These polynomials for some infinite classes of fullerenes and nanotubes are also computed.